It's important to note that there's no such thing as a truly "objective" position in Google, or any search engine. You should always expect some discrepancy due to factors such as:
Volatility of ranking results;
Time lag between ranking update and manual checking;
Personalisation of search results.
Volatility of ranking results
Our research has shown that search listings after the first page of Google tend to fluctuate a lot more often than those on the front page.
So while discrepancies found for the top 5 results should be very small, results at #10-#20 are a lot more volatile and can easily differ by 5 positions each update.
Time lag between ranking update and manual checking
Google search results are very dynamic. Most likely, anywhere from a few hours to a few days will pass in between the time Ahrefs pulls rankings and the time you manually check the SERPs.
These positions are likely to have experienced some fluctuations by the time you check them, especially if they were outside of the top 5 results.
Personalisation of search results
This includes factors such as:
Location
You'll find that if you ask some friends from different parts of town to check on your ranking position at any given time, they'll often report different numbers. This is because of how Google "personalises" search results by location.
Likewise, Ahrefs is unable to pull search results from the same IP network as you. This accounts for some discrepancy - again, small in the top 5 positions, and larger for other positions.
Search History
Google customises search results for users based on previous searches and clicked links. In other words, each individual's interests and search habits have an effect on search results he is shown.
Social Networks
Google+ data such as your age, interests, gender, social connections and location all contribute to personalising your search results. For example, if you're logged into your Google account, you might be served search results that have previously been shared within your social network.
A couple of points to enhance accuracy
Make sure that you're using "Incognito" ("Private") mode in your browser (so that your browser cookie record does not influence the search results) and you're logged out from your Google account
You can use a VPN to try different IP addresses within the same country or region and check search results, and see how differently they change. Do note you can select a more specific location in a country to track keywords in Rank Tracker
How to check when the latest SERP was fetched for a keyword in Rank Tracker
Hover over the date in the Updated column:
The time and date is in the UTC timezone.